Quantum effects Over the years, many suggested effects have been thought to be characteristic of quantum systems: • Basic quantization (1913): mechanical properties of particles in effectively bounded systems are discrete; • Wave-particle duality (1923): objects like electrons and photons can be described as either waves or particles; • Spin (1925): particles can have intrinsic angular momentum even if they are of zero size; • Non-commuting measurements (1926): one can get different results doing measurements in different orders; • Complex amplitudes (1926): processes are described by complex probability amplitudes; • Probabilism (1926): outcomes are random, though probabilities for them can be computed; • Amplitude superposition (1926): there is a linear superposition principle for probability amplitudes; • State superposition (1926): quantum systems can occur in superpositions of measurable states; • Exclusion principle (1926): amplitudes cancel for fermions like electrons to go in the same state; • Interference (1927): probability amplitudes for particles can interfere, potentially destructively; • Uncertainty principle (1927): quantities like position and momenta have related measurement uncertainties; • Hilbert space (1927): states of systems are represented by vectors of amplitudes rather than individual variables; • Field quantization (1927): only discrete numbers of any particular kind of particle can in effect ever exist; • Quantum tunnelling (1928): particles have amplitudes to go where no classical motion would take them; • Virtual particles (1932): particles can occur for short times without their usual energy-momentum relation; • Spinors (1930s): fermions show rotational invariance under SU(2) rather than SO(3); • Entanglement (1935): separated parts of a system often inevitably behave in irreducibly correlated ways; • Quantum logic (1936): relations between events do not follow ordinary laws of logic; • Path integrals (1941): probabilities for behavior are obtained by summing contributions from many paths; • Imaginary time (1947): statistical mechanics is like quantum mechanics in imaginary time; • Vacuum fluctuations (1948): there are continual random field fluctuations even in the vacuum; • Aharonov–Bohm effect (1959): magnetic fields can affect particles even in regions where they have zero strength; • Bell's inequalities (1964): correlations between events can be larger than in any ordinary probabilistic system; • Anomalies (1969): virtual particles can have effects that violate the original symmetries of a system; • Delayed choice experiments (1978): whether particle or wave features are seen can be determined after an event; • Quantum computing (1980s): there is the potential for fundamental parallelism in computations.

Meanwhile, there continued to be ever more accurate experimental tests of general relativity in the solar system—and at least in the weak gravitational fields available there (with metrics differing from the identity by at most one part in 10 6 ), all have worked out to around the 10 -3 level.

Mathematica is available from Wolfram Research for all standard computer systems; much more information about it can be found on the web, especially from www.wolfram.com . There are many books about Mathematica—the original one being my The Mathematica Book .

So while it remains impossible to work out all the consequences of string theories, it is conceivable that among the representations of such theories there might be ones in which matter can be viewed as just being associated with features of space.

The primary ones at which I pursued projects that helped me in formulating issues for this book were Bell Laboratories, Los Alamos National Laboratory and Thinking Machines Corporation.

. • 1981: I begin to study 1D cellular automata, and generate a small picture analogous to the one of rule 30 on page 27 , but fail to study it. • 1984: I make a detailed study of rule 30, and begin to understand the significance of it and systems like it.